Abelian extensions of an absolutely unramified local field with general residue field
type: Item
instance of: scholarly article
title: Abelian extensions of an absolutely unramified local field with general residue field
zbMATH Open document ID: 0666.12012
DOI: 10.1007/BF01394341
author: Masato Kurihara
published in: Inventiones Mathematicae
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143605
review text: By analyzing the dual group \(H^ 1(K,\mathbb Q/\mathbb Z)\) of the maximal abelian extension of a Henselian discrete valuation field \(K\) in which an odd prime \(p\) serves as a prime element, the feature of abelian extensions of \(K\) is completely determined. Observe that there is no restriction imposed on the residue field \(F\) of \(K\). The main part of the paper deals with the construction of a certain surjective homomorphism \[ \phi_ n: H^ q(K,\mathbb Z/p^ n(q-1))\to W_ n\Omega_ F^{q-1}, \] where \(W_ n\Omega_ F^{\cdot}\) is the de Rham-Witt complex and where \(\mathbb Z/p^ n(q-1)\) means the Tate twist. Restricted to \(q=1\) there results an exact sequence \[ 0\to H^ 1(K,\mathbb Q_ p/\mathbb Z_ p)_{nr}\to H^ 1(K,\mathbb Q_ p/\mathbb Z_ p)\to \lim_{\vec V}W_ n(F)\to 0, \] where \(nr\) abbreviates ``non-ramified'' and \(V\) stands for ``Verschiebung'' of Witt vectors over \(F\). As an additional consequence, the author achieves several results on the Brauer group of \(K\) (compare also [\textit{H. Miki}, J. Fac. Sci., Univ. Tokyo, Sect IA 21, 377--394 (1974; Zbl 0301.12003)]).
Mathematics Subject Classification ID: 11S25
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 12J10
Mathematics Subject Classification ID: 13J15
Mathematics Subject Classification ID: 14F22
Mathematics Subject Classification ID: 14F30
zbMATH DE Number: 4089675
zbMATH Keywords: maximal abelian extension
zbMATH Keywords: Henselian discrete valuation field
reviewed by: Jürgen Ritter
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4089675